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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Critical Thinking

  • question_answer
    If \[I=\int_{{}}^{{}}{{{e}^{x}}\sin 2x\ dx}\], then for what value of K, \[KI={{e}^{x}}(\sin 2x-2\cos 2x)+\]constant             [MP PET 1992]

    A) 1

    B) 3

    C) 5

    D) 7

    Correct Answer: C

    Solution :

    • \[I=\int_{{}}^{{}}{{{e}^{x}}\sin 2x\,dx}=\sin 2x\,.\,{{e}^{x}}-2\int_{{}}^{{}}{\cos 2x\,.\,{{e}^{x}}dx}\]                      
    • \[=\sin 2x\,.\,{{e}^{x}}-2\cos 2x\,.\,{{e}^{x}}-4\int_{{}}^{{}}{{{e}^{x}}\sin 2x\,dx}\]                   
    • \[\Rightarrow 5I={{e}^{x}}(\sin 2x-2\cos 2x)+\]Constant                
    • Equating the given value, we get \[K=5.\]


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